Physical Hierarchies

June 7, 2010

Hierarchies are an important learning aid. Just as datelines help to put historical facts into perspective, ordering objects according to the strength of a physical property allows a better understanding of both the objects and that property. One of the simplest materials hierarchies is the Mohs scale of mineral hardness, which is a ranking of the hardness of common minerals according to which will scratch the others. This hierarchy of materials is shown in the table, along with a more analytical measure of hardness, the Sclerometer hardness.

Moh Hardness

Sclerometer Hardness

Mineral

Composition

1

1

Talc

Mg3Si4O10(OH)2

2

3

Gypsum

CaSO4(H2O)2

3

9

Calcite

CaCO3

4

21

Fluorite

CaF2

5

48

Apatite

Ca5(PO4)3(OH,Cl,F)

6

72

Orthoclase Feldspar

KAlSi3O8

7

100

Quartz

SiO2

8

200

Topaz

Al2SiO4(OH,F)2

9

400

Corundum

Al2O3

10

1600

Diamond

C

One of the most fundamental properties is length, and physical size was likely the most important hierarchy to early humans - Small animals are for eating, but large animals should be avoided. This early hierarchy, which spanned the size scale from grains of sand (~1 mm) to heights of mountains (~10,000 meters), covered seven orders of magnitude. Scientific instruments have probed length over an incredibley larger scale, from the diameter of the electron (5.63x10-15 meters) to the diameter of the observable universe (7.37x 1023 meters).

Hierarchies are easy to build when quantities have numbers attached to them, like length, but hierarchies were built throughout much of history based on what were thought to be obvious qualitative principles. A short article in the May issue of Nature Physics [1] used an illustration, called the Lullian staircase from a 1512 book, De nova logica, by Ramon Llull, as shown below [2].

Although it's hard to read from this small figure, Llull ordered all objects, both physical and spiritual, according to the hierarchy, Earth, Fire, Plants, Animals, Humans, Celestial Objects, Angels, and God. This hierachy seems to proceed according to complexity. As a materials scientist, I would reverse the order of Fire and Earth, since mineral crystals have a much lower entropy than hot air. The ordering of the rest seems logical to me, but in this case I'm thinking more like a human and less like a scientist.